General | Math & Conversions Regular Polygon Calculator Regular Polygon Calculator (Area, Perimeter, Apothem, Angles) Solve for all geometric properties of any regular polygon. Input the **Number of Sides ($n$)** and **one** known dimension (Side Length $s$, Apothem $a$, or Radius $R$). Input (Minimum $n$ and one dimension) Number of Sides ($n$) Side Length ($s$) Apothem ($a$) Radius ($R$) Calculate Polygon Properties Key Results Perimeter ($P$) Apothem ($a$) Radius ($R$) Area ($A$) Angle Measures (Degrees) Internal Angle ($\theta_i$) Central Angle ($\theta_c$) Exterior Angle ($\theta_e$) Angle Sums (Total) Sum of Internal Angles Sum of Exterior Angles Step-by-Step Solution The Fundamental Triangle and Trigonometry The key to solving any regular N-gon is to divide it into $n$ congruent **isosceles triangles**. Each triangle is formed by two radii ($R$) and one side ($s$). Bisecting this triangle creates a **right triangle**, known as the fundamental triangle, with the following properties: **Leg 1:** Apothem ($a$). **Leg 2:** Half of the side length ($s/2$). **Hypotenuse:** Radius ($R$). **Angle at Center:** Half of the central angle ($\frac{180^\circ}{n}$). Using this right triangle and the tangent function, we establish the core relationship: $$\tan\left(\frac{180^\circ}{n}\right) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{s/2}{a}$$ Key Formulas for a Regular Polygon Once one property (like the apothem) is known, all other properties can be calculated. Property Formula Perimeter ($P$) $P = n \cdot s$ Area ($A$) $$A = \frac{1}{2} a P \quad \text{or} \quad A = \frac{1}{4} n s^2 \cot\left(\frac{180^\circ}{n}\right)$$ Internal Angle ($\theta_i$) $$\theta_i = \frac{(n-2) \times 180^\circ}{n}$$ Central Angle ($\theta_c$) $$\theta_c = \frac{360^\circ}{n}$$ Frequently Asked Questions (FAQ) What defines a regular polygon? A regular polygon (or N-gon) is a closed, two-dimensional shape with 'N' number of equal sides and 'N' equal interior angles. Examples include the square (N=4), pentagon (N=5), and hexagon (N=6). What is the formula for the sum of the interior angles? The sum of all interior angles of any N-sided polygon (regular or irregular) is found using the formula: $\\text{Sum} = (n - 2) \times 180^\\circ$. What is the difference between Apothem and Radius? The **Radius ($R$)** is the distance from the center to a vertex, forming the circumscribed circle. The **Apothem ($a$)** is the perpendicular distance from the center to the midpoint of a side, forming the inscribed circle. What is the formula for the area using only the side length ($s$)? The area can be calculated using only the side length ($s$) and the number of sides ($n$) with the cotangent function: $\\text{Area} = \frac{1}{4} n s^2 \cot(\\frac{180^\\circ}{n})$. Key Polygon Formulas Area $$A = \frac{1}{2} a P$$ Internal Angle $$\theta_i = \frac{(n-2)180^\circ}{n}$$ Apothem (from Side $s$) $$a = \frac{s}{2 \tan\left(\frac{180^\circ}{n}\right)}$$ Related Geometry Tools Regular N-gon Calculator (Alternate) Polygon Interior Angles Area Calculator (General) Perimeter Calculator Irregular Polygon Calculator